How Do You Solve a Sequence Summation Problem with Variable k?

[david18]

Homework Statement

A sequence of terms U1, U2, U3, ... is defined by Un = 5n + 20

If the sum of the first N terms of the sequence defined above equals the sum of the first kN natural numbers, show that:

N = 45 - k / k^2 - 5

Homework Equations

Sum of first N natural numbers = N(N+1)/2

Sum of sequence = N(50+(N-1)5)/2 (knowing that a=25, d=5)

The Attempt at a Solution

I put the substituted kN into N in the equation N(N+1)/2 to give me kN(kN+1)/2.

I then made kN(kN+1)/2 = N(50+(N-1)5)/2 and solved from there which eventually got me to:

k^2N^2 + kN = 50N + 5N^2 -5

(I then divided both sides by N as N is always positive)

N = 45 +5N -k / k^2

...the +5N makes it wrong. I don't know where I've gone wrong, any help would be appreciated.

 

[Kurret]

Your last expression is right, the only thing you need to do is solve for N.

 

[kamerling]

Homework Statement

k^2N^2 + kN = 50N + 5N^2 -5

This one should be k^2N^2 + kN = 50N + 5N^2 -5N, but I suppose this was an inconsequential typo.

(I then divided both sides by N as N is always positive)

N = 45 +5N -k / k^2

Dividing by N is OK, but this is not the result. Just divide the above with N (all the terms contain N so this is easy), and then group terms with and without N.